Multi-stability analysis of fractional-order quaternion-valued neural networks with time delay

Abstract

This paper addresses the problem of multi-stability analysis for fractional-order quaternion-valued neural networks (QVNNs) with time delay. Based on the geometrical properties of activation functions and intermediate value theorem, some conditions are derived for the existence of at least $ (2\mathcal{K}_p^R+1)^n, (2\mathcal{K}_p^I+1)^n, (2\mathcal{K}_p^J+1)^n, (2\mathcal{K}_p^K+1)^n $ equilibrium points, in which $ [(\mathcal{K}_p^R+1)]^n, [(\mathcal{K}_p^I+1)]^n, [(\mathcal{K}_p^J+1)]^n, [(\mathcal{K}_p^K+1)]^n $ of them are uniformly stable while the other equilibrium points become unstable. Thus the developed results show that the QVNNs can have more generalized properties than the real-valued neural networks (RVNNs) or complex-valued neural networks (CVNNs). Finally, two simulation results are given to illustrate the effectiveness and validity of our obtained theoretical results.

Keywords

• multiple stability
• fractional-order
• caputo fractional derivative
• quaternion-valued neural networks
• time delay